Sample space variables

In words, the right-hand side is the probability that the random variable U (x, t) falls between the sample space values V) and V) + dV) for different realizations of the turbulent flow.5 In a homogeneous flow, this probability is independent of x, and thus we can write the one-point PDF as only a function of the sample space variable and time /(Vi i ). 

A semi-colon is used in the argument list to remind us that V is an independent (sample space) variable, while x and t are fixed parameters. Some authors refer to fyx (Vj x, f ) as the one-point, one-time velocity PDF. Here we use point to refer to a space-time point in the four-dimensional space (x, t). 

Rules of matrix algebra can be appHed to the manipulation and interpretation of data in this type of matrix format. One of the most basic operations that can be performed is to plot the samples in variable-by-variable plots. When the number of variables is as small as two then it is a simple and familiar matter to constmct and analyze the plot. But if the number of variables exceeds two or three, it is obviously impractical to try to interpret the data using simple bivariate plots. Pattern recognition provides computer tools far superior to bivariate plots for understanding the data stmcture in the //-dimensional vector space. 

A random variable is a real-valued function defined over tlie sample space S of a random experiment (Note tliat tliis application of probability tlieorem to plant and equipment failures, i.e., accidents, requires tliat tlie failure occurs randomly. 

Some physiological variables influence the measurement of fibrinolytic activators and inhibitors. For instance, both t-PA and plasminogen activator inhibitor 1 (PAI-1) levels in plasma are subject to diurnal variation in a 12-hour period. Even in samples taken at the same time of day the coefficient of variation (CV) of measured PAI levels range from 8 to 143% To account for this diurnal variation, blood samples spaced over several time intervals during a 24-hour period should be collected. Consumption of alcohol induces the PAI level in plasma. The half-life of t-PA is 360 seconds. However, in the presence of trauma or inflammation, when the PAI-1 level is expected to be elevated 10-fold, the half-life of t-PA is reduced to 36 seconds (114). 

Alternatively, an LES joint velocity, composition PDF can be defined where both (j> andU are random variables Aj 0 U 4 U >4 x, t). In either case, the sample space fields U and0 are assumed to be known. 

HCA is a common tool that is used to determine the natural grouping of objects, based on their multivariate responses [75]. In PAT, this method can be used to determine natural groupings of samples or variables in a data set. Like the classification methods discussed above, HCA requires the specification of a space and a distance measure. However, unlike those methods, HCA does not involve the development of a classification rule, but rather a linkage rule, as discussed below. For a given problem, the selection of the space (e.g., original x variable space, PC score space) and distance measure (e.g.. Euclidean, Mahalanobis) depends on the specific information that the user wants to extract. For example, for a spectral data set, one can choose PC score space with Mahalanobis distance measure to better reflect separation that originates from both strong and weak spectral effects. 

In statistics, a variable, often denoted by a capital letter, that associates a number with the outcome of a random experiment. More formally, a random variable is a function that assigns a real number to each outcome in the sample space of a random experiment. See Statistics A Primer)... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

Defining a random variable on a sample space S amounts to coding tlie outcomes in real numbers. Consider, for example, the random experiment involving die selection of an item at randoni from a manufactured lot. Associate X = 0 widi die drawing of a non-defective item and X = 1 widi die drawing of a defective item. Tlien X is a randoni variable with range (0, 1) and dierefore discrete. 

A random variable, Z. may be constructed on a sample space by specifying its values Zi, Z2. on the sample points. Such variables are useful in modeling random errors of observations. The mean value of Z over the sample space is then given by the relations... 

Equations (4.3-4) and (4.3-5) are the first of several important limit theorems that establish conditions for asymptotic convergence to normal distributions as the sample space grows large. Such results are known as central limit theorems, because the convergence is strongest when the random variable is near its central (expectation) value. The following two theorems of Lindeberg (1922) illustrate why normal distributions are so widely useful. 

A real, symmetric matrix A is called positive definite if x Ax > 0 for every conforming nonzero real vector x. Extend the result of (a) to show that the covariance matrix E in Eq. (4.C-1) is positive definite if the scalar random variables i ,.... Emu are linearly independent, that is, if there is no nonzero m-vector x such that x Eu vanishes over the sample space of the random vector . 

Figure 13.8 WA plot for the sample of Figure 13.7 Logarithm of the Fourier coefficients of the Oil) profiles of the stabilized zirconia phase for several values of Fourier length (on the right, in nm), plotted as a function of the square of the reciprocal space variable. (Reprinted from ref. 38 with the permission of the International Union of Crystallography.)...

A specific, in some sense the simplest Markov Chain Monte Carlo algorithm (—> Monte Carlo methods). In Gibbs sampling one samples the variables of a solution space one after the other. Each variable is drawn from a conditional distribution on the values of the other variables that were pre-... 

The pore-scale model provides the basis for development of a statistical framework for upscaling from a single pore to a sample of variably saturated porous medium. The statistical distribution of pore sizes is modeled as a gamma distribution with the expected values of liquid configuration in pore space calculated from geometrical and chemical potential considerations within the statistical framework. Model predictions compare favorably with measured retention data, yielding similarly close fits to data as the widely used van Genuchten parametric model. Liquid-vapor interfacial area as a function of chemical potential is readily calculated from estimated retention parameters. Comparisons of calculated inter-... 

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